On generalized Hilbert-Kunz function and multiplicity

2013/05/08 by Dao, Hailong, Smirnov, Ilya · 1 citation
#13A35 (Primary) #13D07 #13H10 (Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.1833

Abstract

Let (R,\mathfrak m) be a local ring of characteristic p>0 and M a finitely generated R-module. In this note we consider the limit: lim_{n→ ∞} \fracℓ(H⁰_{\mathfrak m}(Fⁿ(M)))p^{ndim R} where F(-) is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when R is excellent and has isolated singularity. Furthermore, if R is a complete intersection, then the limit is 0 if and only if the projective dimension of M is less than the Krull dimension of R. We exploit this fact to give a quick proof that if R is a complete intersection of dimension 3, then the Picard group of the punctured spectrum of R is torsion-free. Our results work quite generally for other homological functors and can be used to prove that certain limits recently studied by Brenner exist over projective varieties.

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• Generalized Hilbert-Kunz Multiplicity for Families of Ideals

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